Invariance of characteristic values and L∞ norm under lossless positive real transformations

In this paper the invariance of the characteristic values and of the L ∞ norm of linear time-invariant (LTI) systems under lossless positive real transformations is proven. Given a LTI system with transfer function matrix G ( s ) , the transformation s ← F ( s ) with F ( s ) being an arbitrary lossless positive real function of order nF is considered, and the algebraic Riccati equations (AREs) allowing to assess some properties of the transformed system G ( F ( s ) ) are investigated. It is proven that, under such transformations, the solutions of the AREs associated to system G ( F ( s ) ) are related to those of G ( s ) . From this property, it derives that G ( F ( s ) ) and G ( s ) have the same L ∞ norm and that the characteristic values of G ( F ( s ) ) are those of G ( s ) , each with multiplicity nF.